Exact Parametric Solutions for the Intersections of Quadric Surfaces Using MAPLE

Abstract

Quadric surfaces play a very important role in solid geometric modeling and in the design and fabrication of mechanical and industrial parts. Solving the intersection curve between two quadrics is a fundamental problem in computer graphics and solid modeling. We present a new analytical method for parameterizing the intersection curve of two quadrics, which are represented by implicit quadratic equations in 3D. The method is based on the observation that the intersection curve of two quadrics comprises all the points that satisfy a parametric second order polynomial system. We show that the computation of the intersection problem of two general quadrics can be reduced to the solution of quartic polynomials. In particular, we show that the intersection problem of two quadric surfaces that are expressed in canonical forms can be reduced to the solution of quadratic polynomials. All the exact parametric solutions for the intersections of quadric surfaces are implemented in the Computer Algebra System MAPLE. Several previously published test problems of the intersection of quadric surfaces are presented and discussed.